These are tricky ones, since a literal translation into English produces a phrase used with quite a different meaning.

"Give the word" = "dar la orden" (de atacar, por ejemplo)
"Have the word" not being used as such, it would be more likely to be "have the right word" - "tener la palabra justa" - or "have the word on the tip of one's tongue".

Apart from Adolfo's excellent explanation above, in a less formal situation when people are discussing important issues and may need to be given a turn to speak at some length, "tener la palabra" would be expressed in English as "have one's say" or the person letting them speak might say "over to you".

The only example which translates literally and carries the identical meaning is "dar la palabra a alguien", which in English must have the personal possessive instead of "the", as in Adolfo's example: "le di mi palabra..." = "I gave him my word" = I promised him.